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This paper claims that in a one-dimensional driven-dissipative Bose-Hubbard model, the spectral-function linewidth in the weakly interacting, large-filling regime follows Kardar-Parisi-Zhang scaling, growing as k^z with z=3/2 rather than th

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2026-08-01 02:12 UTC pith:BENTGA2Z

load-bearing objection Genuinely new results and a usable tensor-network workflow; the KPZ linewidth claim is plausible but the spectral collapse is a weaker test than presented because the amplitude exponent is fitted, and the one-loop comparison has a circular fit. the 4 major comments →

arxiv 2607.25508 v1 pith:BENTGA2Z submitted 2026-07-28 cond-mat.quant-gas

From weakly to strongly-interacting driven-dissipative bosons in one dimension

Martina Z\"undel , Lo\"ic Herviou , L\'eonie Canet , Anna Minguzzi This is my paper
classification cond-mat.quant-gas
keywords driven-dissipative Bose-HubbardKardar-Parisi-Zhang universalityspectral functionlinewidth scalingLiouvillian gaptensor networksGross-Pitaevskii equationKeldysh formalism
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper argues that a one-dimensional driven-dissipative Bose-Hubbard model with incoherent pump and one- and two-body losses can be understood through its two-point space-time correlations in two complementary regimes. In the weakly interacting, large-filling regime, phase fluctuations are governed by KPZ universality, and the spectral linewidth scales as k^{3/2} instead of the Bogoliubov k^2, offering a directly measurable signature. In the strongly interacting, low-filling regime, a tensor-network calculation shows that the mean-field threshold is not accompanied by a closing Liouvillian gap; instead, the slowest excitation undergoes a level crossing, changing from a uniform density mode to a staggered single-particle mode. These results connect universal dynamical scaling to experimentally accessible spectral observables and clarify what survives of the mean-field transition in one dimension.

Core claim

The central claim is that in the semi-classical regime the momentum dependence of the spectral linewidth is governed by KPZ scaling, Γ(k) ~ k^z with z=3/2, in contrast to the Bogoliubov prediction z=2. The authors establish this by simulating the stochastic driven-dissipative Gross-Pitaevskii equation at parameters where phase fluctuations dominate, verifying KPZ exponents (β≈1/3, χ≈1/2) and Tracy-Widom statistics, and showing that spectral lines at different momenta collapse when rescaled by k^{3/2}. A second, independent claim is that in the fully quantum model the Liouvillian gap does not close when crossing the mean-field transition point; instead, the longest-lived excitation changes it

What carries the argument

The argument is carried by the mapping from the driven-dissipative Gross-Pitaevskii equation to the KPZ equation for the condensate phase, valid when density fluctuations are negligible so that |g^{(1)}(x,t)| ≈ exp(−C_{θθ}(x,t)/2). This relation turns the KPZ phase-growth scaling into a prediction for the spectral linewidth: Γ(k) ~ k^z with z=3/2. In the quantum regime, the key tool is a matrix-product-operator representation of the Liouvillian, with DMRG-based steady-state search and TEBD/TDVP time evolution to obtain retarded Green's functions, complemented by a one-loop Keldysh self-energy calculation for the weak-interaction linewidth.

Load-bearing premise

The KPZ linewidth claim rests on the assumption that density fluctuations are negligible, so that the first-order correlation function is set entirely by phase fluctuations (Eq. 19), and that the chosen parameters (Eq. 20) place the system inside the KPZ basin, away from vortex and soliton sectors; the paper itself shows the scaling fades when the density drops by 25% or the system size falls below L=2^8.

What would settle it

Measure or simulate the spectral function of a one-dimensional driven-dissipative condensate at the parameters of Eq. (20) with large filling and system size L≥2^8: if the extracted linewidth follows Γ(k) ~ k^2 over a range of k instead of collapsing onto Γ(k) ~ k^{3/2}, the central claim fails. Equivalently, in numerics, reducing the density by 25% at fixed other parameters should destroy the k^{3/2} collapse; observing k^{3/2} even at low density would contradict the claimed parameter window.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The spectral function provides a direct, experimentally accessible observable to detect KPZ scaling in one-dimensional driven-dissipative condensates, such as exciton-polariton systems.
  • Bogoliubov linear-response theory fails to capture the long-time phase dynamics; at low momenta the nonlinear KPZ term dominates and changes the linewidth exponent from 2 to 3/2.
  • In one dimension, the mean-field pump-loss threshold does not correspond to a genuine dissipative phase transition: the steady state varies smoothly, but the slowest relaxation mode changes its nature at a level crossing.
  • One-loop Keldysh perturbation theory quantitatively captures the temporal decay of the retarded Green's function in the weak-interaction, low-filling quantum regime, including the linear-in-γ₂ₗ contribution from two-body losses.
  • KPZ scaling in the linewidth is fragile: it requires sufficiently large systems (L ≥ 2^8) and sufficiently high density (within 25% of the KPZ parameter set), and is lost when vortices or solitons appear.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the spectral-linewidth collapse is confirmed in polariton experiments, it would provide a simpler probe of KPZ universality than full space-time correlation measurements, which are harder to access.
  • The level-crossing mechanism in the quantum regime may be a finite-size precursor of the higher-dimensional dissipative transition, suggesting that in one dimension the transition is replaced by a crossover in excitation structure that could persist at larger U/J.
  • Because the KPZ linewidth is lost when density fluctuations grow, tuning the two-body loss rate may offer a control knob to switch between Bogoliubov and KPZ behavior in the same device.
  • The strong discrepancy between semi-classical and quantum steady-state densities for the same microscopic parameters implies that reaching KPZ scaling in a fully quantum simulation would require much larger fillings and system sizes than currently accessible.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript studies the one-dimensional driven-dissipative Bose-Hubbard model with incoherent pump, one- and two-body losses, and a nonlocal loss term. It combines stochastic semiclassical simulations of the driven-dissipative Gross-Pitaevskii equation, tensor-network calculations of the quantum NESS and real-time dynamics, and one-loop Keldysh perturbation theory. The main claims are: (i) in the weakly interacting, large-filling semiclassical regime, the spectral-function linewidth shows KPZ scaling, Γ(k) ~ k^z with z = 3/2, instead of the Bogoliubov prediction z = 2; (ii) in the strongly interacting, low-filling quantum regime, the Liouvillian gap does not close at the mean-field transition, but the slowest relaxation mode changes identity at a level crossing; and (iii) a one-loop calculation captures the low-density decay rate of the retarded Green's function.

Significance. If established, the spectral-function KPZ collapse would provide a new, experimentally accessible diagnostic of KPZ universality in driven-dissipative bosonic systems and would extend the earlier work on correlation functions. The tensor-network framework for Lindblad steady states and dynamical two-point functions is a useful methodological contribution, especially the use of a positive-semidefinite L†L minimization and block-sparse symmetry exploitation. The study is also valuable for delineating the narrow parameter window in which semiclassical KPZ behavior survives and for showing that the quantum model at the same microscopic parameters sits in a different density regime. However, the central KPZ spectral-linewidth claim is currently supported by a collapse with a fitted amplitude exponent that is inconsistent with the quoted scaling form, and the one-loop comparison is partly a consistency check rather than an independent prediction. These issues affect the strength of the main conclusions.

major comments (4)
  1. [§III C, Eq. (26), Fig. 6, footnote 1] The central evidence for z = 3/2 is the collapse of spectral lines in Fig. 6 using the scaling variable (ω−ω_k)/|k|^z. However, Eq. (26) has a fixed amplitude prefactor |k|^{-(2χ+d+z)} = |k|^{-7/2} in d = 1, while the collapse multiplies by |k|^{1.89} and footnote 1 explicitly states that the amplitude prefactor does not match pure KPZ. Since α is fitted, the collapse in the frequency variable alone is a weaker test of KPZ universality and could in principle be achieved with other values of z and α. The authors should either demonstrate the full scaling form including the amplitude, or provide a quantitative finite-window/finite-size justification for the mismatch. In addition, the linewidth exponents in Fig. 5 are extracted from Lorentzian fits without error bars or fitting ranges; error bars are needed to assess whether the k^{3/2} dependence is distinguishable from k^2 and k^{3/2} ove
  2. [§IV C, Fig. 9] The agreement between the one-loop result and the tensor-network data in Fig. 9 is presented as a successful prediction, but the lattice-spacing parameter ã is determined from the numerically fitted exponential decay of G^R(k=0,t), as stated in the text: 'From the numerically fitted exponential decay of G^R(k=0,t), we determine the lattice spacing ã'. Using the same data to fix the conversion parameter and then comparing with the same data makes the comparison at least partially circular. The authors should present this as a one-parameter consistency check, not as an ab initio prediction, and should quantify the sensitivity to the fitting procedure and to the small system size (L=3, Ns=5). Without an independent determination of ã, Fig. 9 does not validate the one-loop calculation.
  3. [§III B, Fig. 2, Appendix A] The abstract and introduction state the spectral-function KPZ result for the driven-dissipative Bose-Hubbard model, but the evidence is obtained in a semiclassical regime with density ρ0 ≈ 15.2 a^{-1} and U = 10^{-3}. The manuscript itself shows that the KPZ scaling is lost when the density decreases by 25% or when L < 2^8 (Fig. 2), and Appendix A demonstrates that the fully quantum model at the same microscopic parameters has mean filling ⟨n⟩ ≈ 0.2 and lies in a different density regime. Thus the KPZ spectral-linewidth claim is not established for the quantum lattice model, only for the semiclassical large-filling limit. The authors should state this limitation explicitly in the abstract and conclusions, and should avoid wording that implies the result applies to the full model Hamiltonian (2).
  4. [§III A, Eq. (19)] The mapping from |g^(1)| to the phase-phase correlator assumes small density fluctuations (Eq. (19)), and Fig. 1 shows that this holds only in a limited time window. The authors should state the range of times and momenta over which Eq. (19) is used in the spectral-function analysis, and explain how violations of this assumption affect the extracted linewidth exponent. This is important because the spectral function is computed from the full complex field, not from the phase alone, so the KPZ collapse could be contaminated by density fluctuations outside the quoted window.
minor comments (5)
  1. [Fig. 5] The fitted decay points are shown without statistical uncertainties; adding error bars would also let the reader judge the deviation from the k^{3/2} and k^2 lines at small k.
  2. [§IV B, Fig. 8] The statement that the linewidth depends weakly on wavevector is based on two fitting methods (Lorentzian and HWHM), but the scatter between these methods appears larger than the claimed k-dependence. Please quantify this uncertainty.
  3. [Fig. 6] The y-axis label 'S_k(ω) k' is ambiguous; the rescaling by |k|^{1.89} is only described in the caption. Please make the plotted quantity explicit in the axis label.
  4. [§IV E, Fig. 12/13] The level-crossing analysis is based on exact diagonalization for L=4, Ns=3. It would be helpful to state the dependence of the crossing position and overlap curves on L and Ns, since the conclusion that the gap does not close is limited to these sizes.
  5. [General] The notation Δt in Eq. (20) and in figure axes is used both as a time step and as the unit of frequency; please distinguish the numerical time step from the physical time unit τ.

Circularity Check

1 steps flagged

One one-loop comparison is calibrated to the quantity it then 'predicts'; the central KPZ/linewidth and Liouvillian claims remain independent.

specific steps
  1. fitted input called prediction [Section IV C, Eqs. (30)-(31), Fig. 9]
    "From the numerically fitted exponential decay of G^R(k=0,t), we determine the lattice spacing ã, that relates the couplings in the continuum to the couplings in the lattice model as γ̃2l = ã γ2l, Ũ = ã U. The imaginary part of the self-energy of the retarded Green’s function Σ14 is non-zero, as compared to the case, when γ2l=0, at the same order in perturbation theory, and is found to be linear in γ2l. As displayed in Fig. 9, this is in perfect accordance with the behavior of the decay rate extracted from the tensor network calculations at small γ2l."

    The one-loop self-energy components in Eq. (30) are independent of k, so the retarded propagator (31) gives a decay rate at k=0 that is fixed once ã is set. Determining ã from the numerical k=0 decay of G^R therefore makes the one-loop curve pass through the fitting point by construction; presenting this as 'in perfect accordance' overstates the independence of the check. This is peripheral to the central z=3/2 claim, but it is a genuine fitted-input-called-prediction step.

full rationale

The central results are not equivalent to their inputs. The semi-classical KPZ identification is checked directly against stochastic GPE data: Fig. 1 compares -2 log|g(1)| with the phase-phase correlator and tests β≈1/3, χ≈1/2, and Tracy-Widom statistics; Fig. 5 compares independently fitted Lorentzian linewidths to the k^{3/2} power law; and Fig. 6 attempts a fixed z=3/2 collapse. The quantum part is benchmarked against the exactly known NESS when γ2l=0 and against exact diagonalization results. The one-loop comparison in Sec. IV C is partially circular, as detailed in the step above. The Fig. 6 amplitude mismatch (α≈1.89 vs the KPZ prefactor 7/2 quoted in Eq. (26)) is explicitly acknowledged in footnote 1; it weakens the collapse as a universality test but is not a hidden circularity, because the paper does not claim to predict the amplitude. Some cited KPZ work [44-46,48] involves overlapping authors, but the paper's own checks make those citations non-load-bearing. Appendix A's caveat about the different density regimes in quantum vs semiclassical simulations is a limitation, not a circularity. Overall score reflects one peripheral fitted prediction, not a self-referential derivation chain.

Axiom & Free-Parameter Ledger

4 free parameters · 7 axioms · 0 invented entities

The paper rests on standard Lindblad and Keldysh machinery, plus several domain assumptions about parameter regimes and tensor-network truncations. It introduces no new physical entities. The main bookkeeping items are the hand-tuned KPZ parameters, the fitted lattice rescaling ã, and the fitted collapse exponent α.

free parameters (4)
  • Microscopic parameter set for KPZ simulations = γl=0.07, γp=0.0774, γ2l=2.44e-4, γd=1.27, J=0.9, U=1e-3, a=0.224, Δt=0.02
    Eq. (20): hand-chosen, inspired by polariton experiments, to place the system in the KPZ regime; the z=3/2 linewidth result is conditional on this tuning.
  • Lattice-to-continuum rescaling ã = not quoted
    Sec. IV C: fitted from the numerical exponential decay of G^R(k=0,t) and used to convert lattice γ2l and U into continuum couplings for the one-loop comparison.
  • Collapse amplitude exponent α = ≈1.89
    Fig. 6: fitted to collapse the spectral amplitudes; footnote 1 acknowledges it does not match the predicted pure-KPZ prefactor.
  • Local Hilbert space truncation Ns = 3
    Fig. 10: used for the L=64 strong-interaction spectra; convergence is only demonstrated for L=2 in Appendix A.
axioms (7)
  • domain assumption The Markovian Lindblad master equation (2) with local pump and one-/two-body losses is an accurate description of the driven-dissipative Bose-Hubbard system.
    Central starting point; no independent justification of the Markovian bath approximation is given.
  • domain assumption In the semi-classical limit, all terms non-linear in the quantum field except the purely quadratic noise term can be dropped, yielding the driven-dissipative GPE (11).
    Sec. III; this standard reduction implicitly assumes large occupation and small quantum fluctuations.
  • domain assumption Density fluctuations are negligible in the KPZ regime, so |g(1)| ≈ exp(−C_θθ/2).
    Eq. (19) and Sec. III B; the paper verifies the difference is small for the chosen parameters, but this assumption enables the linewidth-to-phase-correlation mapping.
  • domain assumption The one-dimensional KPZ fixed-point exponents β=1/3, χ=1/2, z=3/2 are the exact universal values for this phase dynamics.
    Sec. III A; taken from literature [21-24,26], not derived in this paper.
  • domain assumption A unique NESS exists and can be obtained variationally as the normalized vector minimizing ||L|ρ⟩||², without enforcing positivity during truncation.
    Eq. (27), Sec. IV A; positivity preservation is not guaranteed under finite-bond-dimension truncation of the vectorized density operator.
  • domain assumption Local boson truncation Ns=3 and the chosen bond dimensions give accurate spectral functions in the strong-interaction regime.
    Sec. IV D, Fig. 10; convergence is only demonstrated for L=2 in Appendix A, not for the L=64 spectra.
  • domain assumption The weak-interaction quantum regime is well described by a vacuum state ρ0=0 plus one-loop self-energy corrections.
    Sec. IV C; assumption stated as 'good approximation since in the numerics ρ is very low'.

pith-pipeline@v1.3.0-alltime-deepseek · 17483 in / 14911 out tokens · 147120 ms · 2026-08-01T02:12:45.450099+00:00 · methodology

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read the original abstract

We consider a one-dimensional driven-dissipative Bose-Hubbard model subjected to incoherent pump and one- and two-body losses, and analyze it by studying its two-point space-time correlations. By employing a combination of numerical methods, such as stochastic semi-classical simulations and tensor network methods, as well as perturbation theory within the Keldysh formalism, we characterize the system at varying filling and interaction strength. We present results for two complementary regimes: i) at large filling and weak interactions, where we show that Kardar-Parisi-Zhang scaling is visible in the linewidth of the spectral function and ii) at weak filling and strong interactions, where we analyze what remains of the mean field transition and identify a change of nature of the excitations. Our study covers two important regions of the phase diagram of such a system.

Figures

Figures reproduced from arXiv: 2607.25508 by Anna Minguzzi, L\'eonie Canet, Lo\"ic Herviou, Martina Z\"undel.

Figure 1
Figure 1. Figure 1: shows our results for the first-order space and time correlations as well as the probability distribution of the phase fluctuations for the parameters given in Eq. (20). We base our identification of the KPZ regime for the phase θ of the condensate on the following three observations: 1. The correlation function g (1) is dominated by the phase fluctuations. Hence, phase-density and density-density correlat… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The fading of the KPZ scaling towards small system sizes [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Zoom on real (left panel) and imaginary (right panel) part of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Width of spectral lines extracted from the spectrum shown [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Spectral function from semi-classical simulation, in the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Tensor network prediction for the spectral function using [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Linewidths and amplitudes of the spectral function shown [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Largest real eigenvalues from exact diagonalization as a [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Linewidths and amplitudes of the spectral function shown [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Normalized squared Hilbert–Schmidt overlap [PITH_FULL_IMAGE:figures/full_fig_p010_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Comparison of the predictions of semiclassical and quan [PITH_FULL_IMAGE:figures/full_fig_p011_14.png] view at source ↗

discussion (0)

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Reference graph

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